Efficient Meshfree Methods For Certain Fractional Differential Equations

Fractional differential equations(FDEs)are advantageous in representation of some typical physical phenomena such as memory process,inheritable characteristics and anomalous diffusion,etc.Consequently,FDEs have gained continuously growing attention and various methods have been proposed to analyze this class of problems,for example,the finite element method.Unlike the integer order derivatives,the fractional order derivatives are non-local and then pose considerable difficulty for numerical simulations,especially for multi-dimensional problems.On the other hand,in the particle-based meshfree methods,arbitrary order smooth shape functions can be easily constructed for desirable accuracy.However,meshfree shape functions usually do not have explicit expressions and thus are very difficult and costly for the fractional derivative computation.This thesis aims to develop efficient and accurate meshfree methods through introducing new weak formulations for certain FDEs,including Riemann-Liouville and time-Caputo and space-Laplacian fractional diffusion equations.As for the 1D Riemann-Liouville fractional diffusion equation,the non-local property of fractional derivatives leads to a dense and non-symmetric stiffness matrix in the conventional Galerkin weak form,in contract to the desirable symmetric and banded stiffness matrix structure for the typical diffusion equations.To resolve this issue,a meshfree formulation that preserves the symmetric and banded stiffness matrix characteristics is proposed through introducing a fractional weight function.It is noted that the stiffness part of the present formulation is identical to its counterpart of the conventional integer order diffusion equation and thus the computation for stiffness matrix becomes trivial.The proposed meshfree method reduces to the finite element method by adjusting the support size of meshfree approximation.Within the proposed method,the meshfree shape functions can be easily utilized.Furthermore,the numerical oscillations arising from the conventional non-symmetric stiffness Galerkin weak form are also alleviated by the proposed formulation.Regarding multi-dimensional Riemann-Liouville fractional diffusion equations,a new weak formulation is established by completely transferring the fractional operators to the weight or test function and only first order derivatives are left for the trial function.Accordingly,a Petrov-Galerkin meshfree method is developed where meshfree shape functions are used for the approximation of trial function,and the discretization of weight function is realized by the finite element shape functions in order to further reduce the computational complexity and improve efficiency.The proposed Petrov-Galerkin meshfree method allows a direct and efficient employment of meshfree approximation,and also removes the singular integration issue in the fractional derivative calculation of meshfree shape functions.It turns out that this method is also applicable to nonlinear fractional differential equations via an analysis of the nonlinear fractional Allen-Cahn equation.Besides,an efficient meshfree method is also developed for multi-dimentional time-Caputo and space-Laplacian fractional diffusion equations,which is characterized by the stabilized conforming nodal integration and lumped mass matrix techniques.The effectiveness of the proposed methods are thoroughly examined by a set of benchmark numerical examples.